Consider a function f(x) = x4 + x3 + x2 + x + 1, where x is a positive integer greater than 1. What will be the remainder if f(x5) is divided by f(x) ?

f(x) = x⁴ + x³ + x² + x + 1 Multiply and divide by (x-1) f(x) = x⁴ + x³ + x² + x + 1 X (x - 1) ÷ (x - 1) f(x) = x⁵ + x⁴ + x³ + x² + x - x⁴ - x³ - x² - x - 1 ÷ (x - 1) f(x) = (x⁵ −1) ÷ (x − 1) Now, f(x⁵) = x²⁰ + x¹⁵ + x¹⁰ + x⁵ + 1 = x²⁰ - 1 + 1 + x¹⁵ - 1 + 1 + x¹⁰ - 1 + 1 + x⁵ - 1 + 1 + 1 = (x²⁰ - 1) + (x¹⁵ - 1) + (x¹⁰ - 1) + (x⁵ - 1) + 5 All the terms in the above is expression is divisible be (x⁵ - 1) except 5. Therefore, the remainder will be 5.